If you do not have access to phpmyadmin, you can try creating a file named info.php with the following contents:

<?php
phpinfo();
?>

and loading it to your web space (in your public_html directory would be good). Then access the file at http://<YOURDOMAINNAME>/info.php

You should delete the file when you are done so as not to share the info with the world.

That will not tell you the name of your databases however. If you have the ability to create databases, then you should have access to the MySQL section of your cPanel server. You can look your DB info up there (or as I said before in phpmyadmin).